A235508 - OEIS

%I #22 Jan 14 2014 17:16:47

%S 0,2,2,2,2,2,1,2,2,2,2,3,3,3,2,3,3,4,1,2,5,4,4,2,3,3,6,1,3,5,4,3,4,3,

%T 2,4,4,3,5,3,4,4,3,3,4,4,5,4,1,2,6,1,3,4,3,5,6,1,2,4,2,4,2,1,3,7,3,3,

%U 6,4,5,6,2,3,7,3,5,4,6,4,8,3,5,2,6,4,6

%N Number of ways to write 2*n = p + q with q > 0 such that p, p*(p+1) - prime(p) and prime(q) - q + 1 are all prime.

%C Conjecture: a(n) > 0 for all n > 1.

%H Zhi-Wei Sun, <a href="/A235508/b235508.txt">Table of n, a(n) for n = 1..10000</a>

%e a(7) = 1 since 2*7 = 11 + 3 with 11, 11*12 - prime(11) = 101 and prime(3) - 3 + 1 = 3 all prime.

%e a(19) = 1 since 2*19 = 37 + 1 with 37, 37*38 - prime(37) = 1249 and prime(1) - 1 + 1 = 2 all prime.

%e a(98) = 1 since 2*98 = 11 + 185 with 11, 11*12 - prime(11) = 101 and prime(185) - 185 + 1 = 919 all prime.

%t p[k_]:=PrimeQ[Prime[k](Prime[k]+1)-Prime[Prime[k]]]

%t q[m_]:=PrimeQ[Prime[m]-m+1]

%t a[n_]:=Sum[If[p[k]&&q[2n-Prime[k]],1,0],{k,1,PrimePi[2n-1]}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000040, A234695, A234851, A235189, A235330, A235661, A235681.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Jan 14 2014